Rigid monoidal category

Remarks

Example: the category $\mathbf{FinVect}_k$ of finite-dimensional vector spaces over a field $k$ is rigid. For $V$ both duals can be taken to be the dual space $V^*$, with evaluation and the coevaluation $1 \mapsto \sum_i e_i \otimes e^i$ or its flip. Non-example: the category $\mathbf{Vect}_k$ of all vector spaces is not rigid, because an infinite-dimensional space has no dual. In a rigid category, a morphism $f \colon A \to B$ has a transpose $f^* \colon B^* \to A^*$, and $A \mapsto A^*$ becomes a contravariant functor. In a braided monoidal category, an object has a right dual if and only if it has a left dual, so asking for one kind suffices. Rigid monoidal categories are also called autonomous categories; a symmetric rigid monoidal category is called compact closed.